332 lines
14 KiB
Text
332 lines
14 KiB
Text
#import "@preview/unequivocal-ams:0.1.1": ams-article, theorem, proof
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#import "@preview/wordometer:0.1.3": word-count, total-words
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#show: ams-article.with(
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title: [The Argument for Betting on God and the Possibility of Infinite Suffering],
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bibliography: bibliography("refs.bib"),
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)
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#set cite(style: "institute-of-electrical-and-electronics-engineers")
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#set text(fractions: true)
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#show: word-count.with(exclude: (heading, <wordcount-exclude>, table))
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#align(
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center,
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table(
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columns: (auto, auto),
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[
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Perm: A2V4847
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],
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[
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Word Count: #total-words #footnote[Figure computed programmatically during document compilation. Discounts content in tables and the AI contribution statement.]<wordcount-exclude>
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],
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),
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)
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= Introduction
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The argument for Betting on God says that you should believe in God, regardless
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of other evidence, purely out of self-interest. In this paper, I challenge this
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argument by assessing the premise that believing in a particular God always
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guarantees the greatest expected utility.
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The author's argument for belief in God #cite(supplement: [p. 38],
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<Korman2022-KORLFA>) goes as follows:
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#pad(
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x: 16pt,
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[
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#set par(first-line-indent: 0pt)
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#smallcaps[The Argument for Betting on God]:
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#pad(
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y: -2pt,
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[(BG1) One should always choose the option with the greatest expected utility
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],
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)
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#pad(
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y: -2pt,
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[(BG2) Believing in God has a greater expected utility than not believing in God
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],
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)
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#pad(
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y: -2pt,
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[(BG3) So, you should believe in God
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],
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)
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],
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)
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BG1 should be uncontroversial. If you expect an action to bring you the most
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utility (i.e. be the most useful), why wouldn't you choose to do it?
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To justify BG2, the author uses a so-called "decision matrix" to compute the
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expected utility of each combination of action and possible outcome. The
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possible actions are placed on the rows, and the possible outcomes are placed
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on the columns, except for the last column, which is the calculated expected
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utility. At each intersection of a row and column, we place the utility we gain
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from that combination of action and outcome. The expected utility for a given
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action is computed by multiplying the utility of each action-outcome pair in
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that row by the probability of the corresponding outcome occurring, and summing
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up all of those values.
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Here is the decision matrix the author proposes on #cite(supplement: [p. 38],
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<Korman2022-KORLFA>) which gives the expected utility for believing or not
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believing in God.
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#show table.cell.where(x: 0): strong
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#show table.cell.where(y: 0): strong
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#figure(
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caption: [Author's decision matrix],
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align(
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center,
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table(
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columns: (auto, auto, auto, auto),
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table.header(
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[],
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[God exists ($50%$)],
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[God doesn't exist ($50%$)],
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[Expected utility],
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),
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[ Believe in God ], [$infinity$], [2], [$infinity$],
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[
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Don't believe in God
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],
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[1],
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[3],
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[2],
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),
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),
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)
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Note that the numerical utility values themselves have no meaning, and they are
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meant to be viewed relative to each other. Utility doesn't literally provide an
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empirical measure of "usefulness" or "happiness."
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We assign the various finite utilities as we see fit, based on how much each
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scenario benefits us. In the case where God does exist, and you believed in
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God, then you are rewarded with an eternal afterlife of bliss and pleasure in
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heaven. This reward is infinitely greater than any possible reward on earth, so
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it has a utility of $infinity$.
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So, the expected utility for not believing is $0.5 times 1 + 0.5 times 3 = 2$,
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and the expected utility is $0.5 times infinity + 0.5 times 2 = infinity$. If,
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according to BG1, you should pick the option with greatest expected utility,
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clearly you should choose to believe in God, because the expected utility is
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$infinity$.
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The exact utilities don't matter much, since any finite utility you could gain
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for atheism cannot possibly be greater than the infinite expected utility of
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believing in God. Also, as the author points out on #cite(<Korman2022-KORLFA>,
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supplement: [p. 40]), the exact probabilities don't matter either since
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multiplying them by $infinity$ still results in the expected utility of
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$infinity$.
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I will show that the Argument for Betting on God fails because BG2 fails. In
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section 2, I argue you cannot determine whether or not believing in God has the
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greatest expected utility because the decision matrix approach fails when
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possible outcomes involving infinitely negative utilities are introduced. In
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section 3, I address a possible response to this objection.
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= Possibility of Infinite Suffering
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It is possible there are more gods than just the one that sends you to an
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eternal afterlife for believing? The author partially addresses this in
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#cite(<Korman2022-KORLFA>, supplement: [pp. 43-44]), using the example of Zeus.
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Zeus will only reward those who believe in him with an eternal afterlife of
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pleasure. So, if you believe in the wrong god, you don't go to the afterlife.
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The author concludes either believing in Zeus or the Christian God still has
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expected utilities of $infinity$, while being an atheist does has a finite
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expected utility. Therefore, it is still preferable to believe in _some_ god
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that may grant you an eternal afterlife, although no argument is made for
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_which_ god.
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However, this leaves out the possibility of gods who punish you for some
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reason. For instance, suppose there exists an _Evil God_ who sends anyone who
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believes in any god to hell for eternity, and does nothing to atheists.
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Let us modify our decision matrix to model an outcome where the Evil God
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exists. #pagebreak()
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#[
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#set figure()
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#figure(
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caption: [Possibility of an Evil God],
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table(
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columns: (auto, auto, auto, auto, auto),
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align: center,
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table.header(
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[],
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[Correct god exists ($33.3%$)],
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[No god exists ($33.3%$)],
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[Evil God exists ($33.3%$)],
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[E.U.],
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),
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[ Believe in some God ], [$infinity$], [1], [$-infinity$], [$?$],
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[
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Don't believe in any God
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],
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[2],
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[3],
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[4],
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[4.5],
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),
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)
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]<other-gods-table>
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We've added the new option to our matrix. For the sake of argument, let's say
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each option has an equally likely outcome. Again, the exact probabilities don't
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really matter when we're multiplying them by infinity.
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The utilities are mostly the same as before. Not believing in any god and the
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Evil God existing is now the best case for the atheist since they avoided
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infinite suffering. However, the theist now faces the possibility of the worst
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case of all: eternal punishment for believing in the wrong god. If eternal
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bliss in heaven has a utility of $infinity$, then it follows that we should
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represent eternal punishment in hell with a utility of $-infinity$.
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There is a problem: how do we calculate the expected utility of believing in
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god? We have $0.333 times infinity + 0.333 times 1 + 0.333 times -infinity$.
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What is $infinity - infinity$? A naive answer might be 0, but infinity is not a
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number in the traditional sense. It makes no sense to add or subtract infinite
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values. For instance, try and subtract the total amount of integers
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($infinity$) from the total amount of real numbers (also $infinity$)
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#footnote[Famously, this infinity is "larger" than the infinite number of
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integers in the sense of cardinality (G. Cantor). But subtracting them still
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makes no mathematical or physical sense.]. Clearly, this notion is meaningless
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and we cannot obtain a solution. So, we consider $infinity - infinity$ an
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_indeterminate form_. So, the expected utility is now _undefined_.
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Consider the following argument:
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#pad(
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x: 16pt,
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[
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#set par(first-line-indent: 0pt)
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#smallcaps[The Indeterminate Utilities argument]:
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#pad(
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y: -1pt,
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[(IU1) If the expected utility of believing in god is undefined, then we
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cannot compare the expected utilities of believing in god or not believing
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in god.],
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)
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#pad(
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y: -1pt,
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[(IU2) The expected utility of believing in god is undefined.],
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)
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#pad(
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y: -1pt,
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[(IU3) So, we cannot compare the expected utilities of believing in god or
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not believing in god.
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],
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)
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#pad(
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y: -1pt,
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[(IU4) If we cannot compare the expected utilities of believing in god or
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not believing in god, then we cannot determine if believing in god has a
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higher expected utility than not believing in god.
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],
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)
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#pad(
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y: -1pt,
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[(IU5) So, we cannot determine if believing in god has a higher expected
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utility than not believing in god. ],
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)],
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)<wordcount-exclude>
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We just showed why the premise IU2 is true, and the conclusion IU5 is in direct
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contradiction with BG2. So, if IU5 holds, then BG2 must fail.
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It's important to note that the Indeterminate Utilities argument doesn't say
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that the _opposite_ of BG2 is true. It doesn't argue that the expected utility
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of being an atheist is greater. In fact, it doesn't say anything about the
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expected utilities, except that they cannot be compared. If they can't be
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compared, then we can't say for certain which option has the higher expected
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utility. Since BG2 claims that believing in god must have the higher expected
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utility, it is a false premise.
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= Addressing Objections
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// == Believing in a god is still preferable to atheism
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//
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// One might argue that believing in a god that rewards believers is always
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// preferable to atheism since you at least have the _opportunity_ to receive
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// eternity in heaven. Perhaps there exists a god who punishes non-believers with
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// eternal damnation. Then, even without the exact expected utility calculation,
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// it's clear that the expected utility of believing in some god must be higher
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// than believing in none as you stand to gain more. Either as a theist or
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// atheist, you run the risk of eternal punishment, but you only have the
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// opportunity to go to heaven by believing in some god rather than none.
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//
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// Fair, the possibility that you are punished for believing in the wrong god
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// doesn't imply that you should be an atheist either. Indeed, there may be a god
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// that punishes atheists. However, there could also exist a god who sends
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// everyone to heaven regardless. Or perhaps they only send atheists to heaven.
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// Either way, there is also the possibility of attaining the infinite afterlife
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// in heaven by being an atheist, so it's still impossible to say that the
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// expected utility of believing in god is must be higher.
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== The Evil God is not plausible
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One might argue that it is not plausible there is an Evil God who punishes all
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theists, including their own believers. Many religions present a god that
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rewards believers and at most punishes disbelievers. None of the major world
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religions propose an Evil God who punishes all believers. It's much more likely
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that a benevolent god exists than an evil one.
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I contend that it doesn't matter whether or not the Evil God is less plausible
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than a benevolent god. Surely, if a rational atheist who is unconvinced by all
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the world's scriptures can still concede that there is at least a non-zero
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chance that some god exists, the rational theist should also concede that there
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is a non-zero chance that the Evil God exists. All it takes is that non-zero
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chance, no matter how small, because multiplying it by $-infinity$ still
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results in the undefined expected utility.
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== Finite utilities
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One might argue that we can avoid using $infinity$ to ensure that all expected
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utility calculations are defined. Instead, suppose that the utility of going to
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heaven is just an immensely large finite number. The utility of going to hell
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is likewise a very negative number. All of our expected utility calculations
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will be defined, and given sufficiently large utilities, we should be able to
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make a similar argument for believing in god.
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// The problem with this argument is that we now open our expected utilities up to
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// individual subjective determination. A core feature of the previous argument
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// involving infinite utilities is that they can effectively bypass numerical
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// comparison. If, instead, finite utilities were used, then each person may
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// assign different utilities to each possible outcome based on their own beliefs.
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// Also, the probabilities are no longer irrelevant, so they must be analyzed as
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// well. This greatly complicates the decision matrix.
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The problem with this argument is that infinity has a special property argument
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relies on. Namely, any number multiplied by $infinity$ is still $infinity$, so
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the exact probabilities we set for the existence of God don't matter. This is
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important for defending against the objection the author mentions on
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#cite(<Korman2022-KORLFA>, supplement: [p. 40]), that the probabilities are
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possibly incorrect, since the numbers don't matter anyways.
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If, instead, only finite utilities were used, then the theist must contend with
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the concern that the probabilities in the matrix are wrong. There could
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conceivably exist a matrix with probabilities for a benevolent god and an Evil
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God such that the expected utility of atheism is actually higher. The issue is
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we cannot say for sure what the probabilities of the benevolent god and the
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Evil God existing are. If we cannot know what the actual probabilities are,
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then we cannot know the final outcome of our matrix. So, without knowing the
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final outcome of the matrix, we still cannot determine whether or not believing
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in god has greater expected utility, and BG2 still fails.
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#pagebreak()
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#[
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= AI Contribution Statement
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#quote[I did not use AI whatsoever in the writing of this paper.]
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]<wordcount-exclude>
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